Table Of ContentAdvanced Structured Materials
Vladimir I. Erofeev
Igor S. Pavlov
Structural
Modeling
of Metamaterials
Advanced Structured Materials
Volume 144
Series Editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke
University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance ofmaterials can beincreasedby combiningdifferent materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers
possible applications in quite diverse fields.
Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc.) and applications.
The topics of Advanced Structured Materials include but are not limited to
(cid:129) classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced
plastics)
(cid:129) metal matrix composites (MMCs)
(cid:129) micro porous composites
(cid:129) micro channel materials
(cid:129) multilayered materials
(cid:129) cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere
structures)
(cid:129) porous materials
(cid:129) truss structures
(cid:129) nanocomposite materials
(cid:129) biomaterials
(cid:129) nanoporous metals
(cid:129) concrete
(cid:129) coated materials
(cid:129) smart materials
Advanced Structured Materials is indexed in Google Scholar and Scopus.
More information about this series at http://www.springer.com/series/8611
Vladimir I. Erofeev Igor S. Pavlov
(cid:129)
Structural Modeling
of Metamaterials
123
Vladimir I.Erofeev Igor S. Pavlov
MechanicalEngineering Research Institute MechanicalEngineering Research Institute
of the Russian Academy of Sciences of the Russian Academy of Sciences
Federal ResearchCenter ‘Institute of Federal ResearchCenter ‘Institute of
AppliedPhysics of the Russian Academy AppliedPhysics of the Russian Academy
of Sciences’ of Sciences’
Nizhny Novgorod,Russia Nizhny Novgorod,Russia
ISSN 1869-8433 ISSN 1869-8441 (electronic)
AdvancedStructured Materials
ISBN978-3-030-60329-8 ISBN978-3-030-60330-4 (eBook)
https://doi.org/10.1007/978-3-030-60330-4
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Preface
At present, technologies of creating advanced structural materials with micro- and
nanostructure are intensively developed. One example of such materials is meta-
materials—anewclassofsubstanceswithacomplexlyorganizedinternalstructure
(microstructure) and possessing unique physical and mechanical properties [1].
They first appeared in the field of optics and photonics [2], but now they are
increasingly found in other areas. For example, acoustic metamaterials are widely
used [3–7]. In particular, they are applied as acoustic absorbers [8]. In addition,
among porous media, granular materials, polymers, composites, and crystalline
media,therearematerialswithanegativePoisson’sratio(auxeticmaterials)[9–17].
However, the creation of metamaterials is extremely difficult without adequate
mathematical models.
In this monograph, the method of structural modeling is proposed to use for
constructing mathematicalmodels ofmetamaterials.Thismethodenablesoneboth
revealingthequalitativeeffectoftheinternalstructureofamaterialonitseffective
elastic moduli and performing quantitative estimates of the moduli. Results of the
performed research can be used for the design of advanced metamaterials with
predetermined physical and mechanical properties.
Thisbook hasbeen written onthebasis ofstudiescarriedoutover thepast two
decades in Mechanical Engineering Research Institute of the Russian Academy of
Sciences(NizhnyNovgorod,Russia),whichisabranchoftheFederalStateBudget
Scientific Institution “Federal Research Center Institute of Applied Physics of the
Russian Academy of Sciences” since 2016.
We are very grateful to our colleagues who, unfortunately, have already passed
away:
to Prof. Alexander Ivanovich Potapov (1949–2010), who was the founder of this
scientific direction at our institute;
toProf.NadezhdaEvgenievnaNikitina(1951–2016),whowasanacoustoelasticity
specialist and made a significant contribution to obtaining the results of Chap. 7,
where prestressed media are considered;
v
vi Preface
to Prof. Gerard A. Maugin (1944–2016) from Pierre and Marie Curie University
(French: Université Pierre-et-Marie-Curie, Paris, France), in collaboration with
whom there were obtained scientific results presented in Chaps. 3 and 4;
to Prof. Alexander Vasilievich Vikulin (1947–2017) from the Institute of
Volcanology and Seismology, Far Eastern Branch of the Russian Academy of
Sciences (Petropavlovsk-Kamchatsky, Russia), who was a specialist in geody-
namics and seismology—discussions with him helped us make the transition from
nanomaterials to geomedia;
to Prof. Leonid lsaakovich Manevitch (1938–2020) from N. N. Semenov Federal
Research Center for Chemical Physics, Russian Academy of Sciences (Moscow,
Russia), who was a specialist in nonlinear dynamics and materials science—
collaboration with him gave us new ideas to choose materials for elaboration
of their models.
We consider as a pleasant duty to thank our colleagues, in co-authorship with
whom the most important results of the monograph were obtained: Professor
I. V. Miloserdova (Nizhny Novgorod Technical State University n.a. R. E.
Alekseev,NizhnyNovgorod,Russia);DoctorsV.V.Kazhaev,A.V.Leontyeva,and
A. O. Malkhanov (Mechanical Engineering Research Institute of the Russian
AcademyofSciences,NizhnyNovgorod,Russia);Prof.A.V.Porubov(Instituteof
Problems of Mechanical Engineering of the Russian Academy of Sciences,
St.Petersburg,Russia);andDr.A.A.Vasiliev(TverStateUniversity,Tver,Russia).
We are grateful to Academician of the Russian Academy of Sciences
V. P. Matveenko; Corresponding Members of the Russian Academy of Sciences
D.A. Indeytsevand A. N.Morozov; Foreign Member of theRussian Academy of
Sciences H. Altenbach; Profs. I. V. Andrianov, S. A. Lurie, A. V. Metrikine,
W. Muller, V. M. Sadovsky, I. N. Shardakov, V. S. Shorkin, and
D. V. Tarlakovsky, for useful scientific discussions and recommendations for
improvement of our book.
We also thank the staff of Mechanical Engineering Research Institute of the
Russian Academy of Sciences: Profs. V. N. Perevezentsev, S. I. Gerasimov,
B. A. Gordeev, V. V. Mishakin, V. M. Rodyushkin, and G. F. Sarafanov for their
attention to our work.
We would like to express our special gratitude to Anastasia Demareva and
Vladimir Sadovsky, postgraduate students of Lobachevsky State University of
Nizhny Novgorod, and to Anna Muravieva, a student of Lobachevsky State
University of Nizhny Novgorod, for their contribution to the development of a
three-dimensional model of a granular medium presented in Chap. 6 of this
monograph.
Nizhny Novgorod, Russia Vladimir I. Erofeev
Igor S. Pavlov
Lobachevsky State University
of Nizhny Novgorod
Preface vii
.
References
1. Gulyaev,Yu.V.,Lagar’kov,A.N.,Nikitov,S.A.:Metamaterials:basicresearchandpotential
applications.HeraldRuss.Acad.Sci.78,268–278(2008)
2. Zhu,S.,Zhang,X.:Metamaterials:artificialmaterialsbeyondnature.Natl.Sci.Rev.5(2),131
(2018)
3. Bobrovnitskii,YuI:Anacousticmetamaterialwithunusualwaveproperties.Acoust.Phys.60
(4),371–378(2014)
4. Bobrovnitskii,YuI:Modelsandgeneralwavepropertiesoftwo-dimensionalacousticmeta-
materialsandmedia.Acoust.Phys.61(3),255–264(2015)
5. Cummer,S.A.,Christensen,J.,Alù,A.:Controllingsoundwithacousticmetamaterials.Nat.
Rev.Mater.1,16001(2016)
6. Fedotovskii, V.S.:Aporousmedium asanacoustic metamaterial withnegative inertialand
elasticproperties.Acoust.Phys.64(5),548–554(2018)
7. Zhou,L.,Jiang,H.:Auxeticcompositesmadeof3Dtextilestructureandpolyurethanefoam.
Phys.StatusSolidiB253(7),1331–1341(2016)
8. Bobrovnitskii,Yu.I.,Tomilina,T.M.:Soundabsorptionandmetamaterials:areview.Acoust.
Phys.64(5),519–526(2018)
9. Engelbrecht J.K., Fridman V.E., Pelinovsky E.N.: Nonlinear Evolution Equations. Pitman,
London.(1988)
10. Evans,K.E.:Auxeticpolymers:anewrangeofmaterials.EndeavourNewSer.4,170–174
(1991)
11. GoldsteinR.V.,GorodtsovV.A.,LisovenkoD.S.Auxeticmechanicsofcrystallinematerials.
MechanicsofSolids.45(4),529–545(2010)
12. Goldstein, R.V., Gorodtsov, V.A., Lisovenko, D.S.: Average Poisson’s ratio for crystals.
HexagonalAuxetics.LettersonMaterials.3(1),7–11(2013)
13. Goldstein,R.V.,Gorodtsov,V.A.,Lisovenko,D.S.:Classificationofcubicauxetics.Physica
StatusSolidiB.250(10),2038–2043(2013)
14. Goldstein R.V., Gorodtsov V.A., Lisovenko D.S.: Cubic auxetics. Doklady Physics. 56(7),
399–402(2011)
15. GoldsteinR.V.,GorodtsovV.A.,LisovenkoD.S.Longitudinalelastictensionoftwo-layered
plates from isotropic auxetics–nonauxetics and cubic crystals. European J. of Mech. – A:
Solids.63,122–127(2017)
16. Goldstein, R.V., Gorodtsov, V.A., Lisovenko, D.S.: The elastic properties of hexagonal
auxeticsunderpressure.Phys.StatusSolidiB253(7),1261–1269(2016)
17. Koniok, D.A., Voitsekhovsky, K.V., Pleskachevsky, Yu.M., Shilko, S.V.: Materials with
negativePoisson’sratio(Thereview).CompositeMech.Des.10,35–69(2004)
Introduction
Predictionofphysical and mechanical propertiesofmediawith microstructureand
adequatedescriptionofdynamic(wave)processesnecessitatemathematicalmodels
taking into account the presence of several scales (structural levels) in a medium,
theirself-consistentinteraction,andthepossibilityofenergytransferfromonelevel
to another. The following scales are usually distinguished [1, 2]: atomic or mi-
croscopic level (characteristic sizes are angstroms and nanometers), mesoscopic
level (from 10−8 to 10−6 m), submacroscopic level (from 10−6 to 10−4 m), and
macroscopic level (over 10−4 m).
Mentalbreakingofamaterialintopartsisrestrictedbysomelimitconsistingina
qualitative change of physical properties on a given scale level; i.e., in this case, a
size effect [3, 4] arises. During studying of wave processes in materials, the size
effectsstarttobeshown,whenthecharacteristicspatialscaleofeffect(e.g., length
of an elastic or electromagnetic wave) becomes comparable with the characteristic
spatial scale of a material—the size of grain, the lattice period, etc.
In process of accumulation of knowledge about microstructure of a material,
therearisesatransitiontonewlevelofknowledge—atheoryiscreatedthatenables
one to explain mechanical behavior of a material from new positions. It should be
emphasized that the actual values of the “microstructure” of the medium in a
specificproblemcanliebothintherangeofnano-ormicrometersandinthelarger
scales.However,fromtheviewpointofthemethodologyoftheoreticalresearch,the
absolute values of the “microstructure” are not so important, as the smallness of
some scales with respect to others.
Frequently, different physical properties of the medium are manifested at dif-
ferent scales. For example, it concerns media such as rocks, particularly, hydro-
carbon reservoirs. The internal structure of the rocks determines at different scales
notonlyvariouselasticproperties,butalsophysicalpropertiessuchasthermaland
electrical conductivity, and hydraulic and dielectric constant [5, 6].
Inthemathematicalsimulationofmicrostructuredmedia,twoapproachescanbe
distinguished:“frommicrotomeso”and“frommacrotomeso.”Thefirstapproach
consistsinthepassagefromatomic-levelmodelstomesoscalemodelsandisbased
onthelawsofquantumtheory.Inthiscase,themediumisconsideredasadiscrete
ix
x Introduction
system of particles coupled by the interaction forces determined from the first
principles(quantumpostulates).Thisapproachallowsonetounderstandthenature
of physical laws and to explain the origin of some properties having no substan-
tiation in the classical theory.
Until the middle of twentieth century, the quantum mechanics was considered,
basically,asthemicroworldmechanics.Beingconstructedonthebasisofquantum
postulates, it does not appear at all on the macroscales, where the continuum
mechanics is valid, which is created on the basis of the laws of conservation of
mass, momentum, kinetic momentum, and the thermodynamics laws (the macro-
scopicfirstprinciples).Thefirstfundamentalstepofthequantummechanicsinthe
field of macroscopic phenomena was the creation of the hydrodynamic theory of
superfluidity of helium-II by L. D. Landau in 1941 and the idea of L. Onsager
(1948)toquantizevortexmotioninit[7].Thenextstepinthisdirectionwasmade
byA.F.AndreevandI.M.Lifshits,whodevelopedin1969thephenomenological
theory of defects in quantum crystals [8]. According to this theory, defects are
considered as delocalized excitations(defectons) that move дeфeктыalmost freely
through the crystal. A crystal with defectons is neither a liquid nor a solid. Two
different types of motion are possible in it. The first type of motion is associated
withsmallvibrationsofthelatticesitesneartheequilibriumstatesandisdescribed
by the classical equations of elastic solid mechanics. The second type is charac-
teristic for a liquid and is associated with quantum diffusion that leads to mass
transfer by defectons, when lattice sites are fixed. At present, such studies are the
subject of quantum macrophysics [9].
Thesecondapproachtomodelingofmicrostructuredmediameanspassingfrom
descriptionofamediumonamacroleveltomesoscalemodels.Withinitsscope,the
elaborationofmathematicalmodelsofsuchmediaproceedsinthreedirections.The
first of them—the continuum-phenomenological direction—is associated with the
construction of generalized continuum models (generalized continua) of the
mechanics of a deformable solid and is based on the classical physics laws. It
involves expanding of the concept of a representative volume of the medium and
taking into account the rotational degrees of freedom of microparticles (polarity
ofthematerial),aswellasaffinedeformationsofthemesovolumeandnon-locality
ofthematerial[10–11].Polarityindicatesthatrigidrotationisallowed,whichisnot
related to the field of displacements in the general case, whereas non-locality tes-
tifies the dependence of the physical properties of the material on the influence of
environmental particles. Continual theories are elaborated by the deductive way:
All the results are consequences of a system offundamental assumptions—axioms
or postulates. The advantages of this elaboration are logical consistency, a rigor
ofthederivationofvariousparticularversionsofthemodels,andthepossibilityof
a consistent classification of theories according to selected attributes. A decisive
contribution to the development of this direction was made by the works of
E. and F. Cosserat [12], C. Truesdell and R. Toupin [13, 14], E. L. Aero and
E. V. Kuvshinskii [15, 16], R. Mindlin [17], A. C. Eringen [18–21], W. Nowacky
[22],V.A.Palmov[23,24],L.I.Sedov[25–27],V.I.Erofeev[28],A.I.Potapov
[29],V.P.Matveenko,I.N.Shardakov,M.A.Kulesh,andE.F.Grekova[30,31],
